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Statistics of Anatomic Geometry Stephen Pizer, Kenan Professor Medical Image Display & Analysis Group University of North Carolina This tutorial and other relevant papers can be found at website: midag.cs.unc.edu Faculty: me, Ian Dryden, P. Thomas Fletcher, Xavier Pennec, Sarang Joshi, Carole Twining MIDAG@UNC Geometry of Objects in Populations via representations z Uses for probability density p(z) Sampling p(z) to communicate anatomic variability in atlases Issue: geometric propriety of samples? Log prior in posterior optimizing deformable model segmentation = registration Optimizez p(z|I), so log p(z) + log p(I|z) Or E(z|I) MIDAG@UNC Geometry of Objects in Populations via representations z Uses for probability density p(z) Compare Medical two populations science Hypothesis testing with null hypothesis p(z|healthy) = p(z|diseased) If null hypothesis is not accepted, find localities where probability densities differ and characterization of shape difference Diagnostic: Is particular patient’s geometry diseased? p(z|healthy, I) vs. p(z|diseased, I) MIDAG@UNC Needs of Geometric Representation z & Probability Representation p(z) Accurate p(z) estimation with limited samples, i.e., beat High Dimension Low Sample Size (HDLSS: many features, few training cases) Measure of predictive strength of representation and statistics [Muller]: d zˆ , z test k 2 k / d z training 2 k test k ,z training 2 where “^” indicates projection onto training data principal space Primitives’ positional correspondence; cases alignment Easy fit of z to each training segmentation or image MIDAG@UNC Needs of Geometric Representation z & Probability Representation p(z) Make significant geometric effects intuitive Null probabilities for geometrically illegal objects Localization Handle multiple objects and interstitial regions Speed and space MIDAG@UNC Schedule of Tutorial Object representations (Pizer) PCA, ICA, hypothesis testing, landmark statistics, objectrelative intensity statistics (Dryden) Statistics on Riemannian manfolds, of m-reps & diffusion tensors, maintaining geometric propriety (Fletcher) Statistics on Riemannian manfolds: extensions and applications (Pennec) Statistics on diffeomorphisms, groupwise registration, hypothesis testing on Riemannian manifolds (Joshi) Information theoretic measures on anatomy, correspondence, ASM, AAM (Twining) Multi-object statistics & segmentation (Pizer) MIDAG@UNC Representations z of Deformation Landmarks Boundary of objects (b-reps) Points spaced along boundary or Coefficients of expansion in basis functions or Function in 3D with level set as object boundary Deformation velocity seq. per voxel Medial representation of objects’ interiors (m-reps) MIDAG@UNC Landmarks as Representation z z = (p1, p2, …,pN) First historically Kendall, Bookstein, Dryden & Mardia, Joshi Landmarks defined by special properties Won’t find many accurately in 3D Global Alignment via minimization of inter-case Spoints distances2 MIDAG@UNC B-reps as Representation z Point samples: z = (p1, p2, …,pN) Like landmarks; popular Characterization of local translations of shell Fit to training objects pretty easy Handles multi-object complexes Global Positional correspondence of primitives Slow reparametrization optimizing p(z) tightness Problems with geometrically improper fits Mesh by adding sample neighbors list Point, normal samples: z = ([p1,n1],…,[pN,nN]) Easier to avoid geometrically improper fits MIDAG@UNC B-reps as Representation z Basis function coefficients z = (a1, a2, …,aM) with p(u) = Sk=1M ak k(u) Achieves geometric propriety Fitting to data well worked out and programmed Implicit, questionable positional correspondence Global, Unintuitive Alignment via first ellipsoid 1 7 12 Representations via spherical harmonics MIDAG@UNC B-rep via F(x)’s level set: z = F, an image Allows topological variability Topology change Global Unintuitive, costly in space Fit to training cases easy: F = signed distance to boundary Modification by geometry limited diffusion Requires nonlinear statistics: not yet well developed Serious problems of geometric propriety if stats on F; needs stats on PDE for nonlinear diffusion Correspondence? Localization: via spatially varying PDE parameters?? MIDAG@UNC Deformation velocity sequence for each voxel as representation z z = ([v1(i.j), v2(i.j),…,vT(i.j)], (i.j) pixels) Miller, Christensen, Joshi Labels in reference move with deformation Series of local interactions Deformation energy minimization Fluid flow; pretty slow Costly in space Slow and unsure to fit to training cases if change from atlas is large MIDAG@UNC M-reps as Representation z Represent the Egg, not the Eggshell The eggshell: object boundary primitives The egg: m-reps: object interior primitives Poor for object that is tube, slab mix Handles multifigure objects and multiobject complexes Interstitial space?? MIDAG@UNC A deformable model of the object interior: the m-rep Object interior primitives: medial atoms Local displacement, bending/twisting, swelling: intuitive Neighbor geometry Objects, figures, atoms, voxels Object-relative coordinates Geometric impropriety: math check MIDAG@UNC Medial atom as a nonlinear geometric transformation Medial atoms carry position, width, 2 orientations deformation T 3 × + × S2 × S2 (× + for edge atoms) From reference atom Hub translation × Spoke magnification in common × Spoke1 rotation × Spoke2 rotation (× crest sharpness) Local M-rep is n-tuple of medial atoms Tn medial atom edge medial atom , n local T’s, a curved, symmetric space Geodesic distance between atoms Nonlinear statistics are required MIDAG@UNC Fitting m-reps into training binaries Optimization penalties Distance between m-rep and binary image boundaries Irregularity penalty: deviation of each atom from geodesic average of its neighbors Yields correspondence(?) Avoids geometric impropriety(?) Interpenetration avoidance Alignment via minimization of inter-case Satoms geodesic distances2 MIDAG@UNC Schedule of Tutorial Object representations (Pizer) PCA, ICA, hypothesis testing, landmark statistics, objectrelative intensity statistics (Dryden) Statistics on Riemannian manfolds, of m-reps & diffusion tensors, maintaining geometric propriety (Fletcher) Statistics on Riemannian manfolds: extensions and applications (Pennec) Statistics on diffeomorphisms, groupwise registration, hypothesis testing on Riemannian manifolds (Joshi) Information theoretic measures on anatomy, correspondence, ASM, AAM (Twining) Multi-object statistics & segmentation (Pizer) MIDAG@UNC Multi-Object Statistics Need both Object statistics Inter-object relation statistics We choose m-reps because of effectiveness in expressing interobject geometry Medial atoms as transformations of each other Relative positions of boundary Spokes as normals Object-relative coordinates MIDAG@UNC Statistics at Any Scale Level Global: z By object z1k Object By figure (atom mesh) z2k Figure neighbors N(z3k) By voxel or boundary vertex Voxel neighbors N(z2k) By atom (interior section) z3k Atom neighbors N(z1k) neighbors N(z4k) Designed for HDLSS atom level voxel level quad-mesh neighbor relations MIDAG@UNC Multiscale models of spatial parcelations Finer parcellation zj as j increases (scale decreases) Fuzzy edged apertures zjk, with fuzz (tolerance) decreasing as j increases Geometric representation zjk We use m-reps to represent objects at moderate scale and diffeomorphisms to modify that representation at small scale Level sets of pseudo-distance functions can represent the variable topology interstitial regions Provides localization MIDAG@UNC Statistics of each entity in relation to its neighbors at its scale level on estimating p(zjk , {zjn: n k}), via probabilities that reflect both interobject (region) geometric relationship and object themselves (also for figures) Focus Markov random field Conditional probabilities p(zjk | {zjn: n k}) p(zjk | {zjn: N(zjk)}) Iterative Conditional Modes – convergence joint mode of p(zjk , {zjn: n k} | Image) = to MIDAG@UNC Representation of multiple objects via residues from neighbor prediction Inter-entity and inter-scale relation by removal of conditional mean of entity on prediction of its neighbors, then probability density on residue | {zjn: N(zjk)}) = p(zjk interpoland zjk: from N(zjk)}) p(zjk Restriction of zjk to its shape space Early coarse-to-fine posterior optimization segmentation results successful, but still under study Alternative to be explored Canonical correlation MIDAG@UNC Want more info? This tutorial, many papers on b-reps, m-reps, diffeomorphism-reps and their statistics and applications can be found at website http://midag.cs.unc.edu 12 MIDAG@UNC MIDAG@UNC